Something happening "specifically after a period" can hint at BV, though what usually happens after the period is over is a smell. It could also be you reacting to the menstrual cup, possibly they have changed the formulation of the plastic? (Though if it were an allergy, seemingly you'd react when you begin to use it, while still on your period.)
Have you been to the ob/gyn lately? It might be a good idea to get swabbed for BV even if you don't fit the classic symptoms.